For atoms trapped under a one-dimensional optical lattice
with the laser frequency \(\omega \)  and the linearly polarized laser field
intensity \(I\) , the Stark shift for a clock transition can be expressed as [1]}, [2]}
\(& h\Delta \nu =[\frac{\partial \Delta \alpha ^{E1}(\omega )}{\partial \nu }\delta -\Delta \alpha ^{QM}(\omega )](n_{z}+\frac{1}{2})\sqrt{\frac{E_{R}}{\alpha ^{E1}(\omega )}}I^{1/2} \nonumber \\ &-[\frac{\partial \Delta \alpha ^{E1}(\omega )}{\partial \nu }+\frac{3}{8}\frac{E_{R}\Delta \gamma ^{l}(\omega )}{\alpha ^{E1}(\omega )}(n_{z}^2+n_{z}+\frac{1}{2})]I \nonumber \\ &+\frac{1}{2}\Delta \gamma ^{l}(\omega )\sqrt{\frac{E_{R}}{\alpha ^{E1}(\omega )}}(n_{z}+\frac{1}{2})I^{3/2}-\frac{1}{4}\Delta \gamma ^{l}(\omega )I^{2},\) 
